in Chronology, the excesses of the solar month above the lunar synodical month, and of the solar year above the lunar year of twelve synodical months; or of several solar months above as many synodical months, and several solar years above as many dozen of synodical months.

The epacts, then, are either annual or menstrual.

Menstrual epacts are the excesses of the civil or calendar month above the lunar month. Suppose, e.g., it were new moon on the first day of January; since the lunar month is 29 days 12h. 44' 3"; and the month of January contains 31 days, the menstrual epact is 1 day 11h. 15' 57".

Annual epacts are the excesses of the solar year above the lunar. Hence, as the Julian solar year is 365 days 6h. and the Julian lunar year 354 days 8h. 48' 38", the annual epact will be 10 days 21h. 11' 22"; that is, nearly 11 days. Consequently the epact of 2 years is 22 days; of 3 years, 33 days; or rather 3, since 30 days make an embolismic or intercalary month.

Thus the epact of 4 years is 14 days, and fo of the rest; and thus, every 19th year, the epact becomes 30 or 0; consequently the 20th year the epact is 11 again; and so the cycle of epacts expires with the golden number, or lunar cycle of 19 years, and begins with the same, as in the following table:

| Gold. Numb. | Epacts. | Gold. Numb. | Epacts. | Gold. Numb. | Epacts. | |-------------|---------|-------------|---------|-------------|---------| | 1 | XI | 8 | XXVIII | 14 | IV | | 2 | XXII | 9 | IX | 15 | XV | | 3 | III | 10 | XX | 16 | XXVI | | 4 | XIV | 11 | I | 17 | VIII | | 5 | XXV | 12 | XII | 18 | XIX | | 6 | VI | 13 | XXIII | 19 | XXX | | 7 | XVII | | | | |

Again, as the new moons are the same, that is, as they fall on the same day every 19 years, so the difference between the lunar and solar years is the same every 19 years. And because the said difference is always to be added to the lunar year, in order to adjust or make it equal to the solar year; hence the said difference respectively belonging to each year of the moon's cycle is called the epact of the said year, that is, the number to be added to the said year, to make it equal to the solar year; the word being formed from the Greek ἐπακτον, induco, intercalo.

Upon this mutual respect between the cycle of the moon and the cycle of the epacts, is founded this rule for finding the Julian epact, belonging to any year of the moon's cycle. Multiply the year given of the moon's cycle into 11; and if the product be less than 30, it is the epact sought; if the product be greater than 30, divide it by 30, and the remainder of the dividend is the epact. For instance, I would know the epact for the year 1712, which is the third year of the moon's cycle. Wherefore 3 is the epact for 1712; for 11 × 3 = 33, and 33 being divided by 30, there is left 3 of the dividend for the epact. But the difference of the Julian and Gregorian years being equal to the excels of the solar above the lunar year, or 11 days, it happens that the Gregorian epact for one year is the same with the Julian epact for the preceding year.